A hierarchical research by large-scale and ab initio electronic structure theories -- Si and Ge cleavage and stepped (111)-2x1 surfaces --

The ab initio calculation with the density functional theory and plane-wave bases is carried out for stepped Si(111)-2x1 surfaces that were predicted in a cleavage simulation by the large-scale (order-N) electronic structure theory (T. Hoshi, Y. Iguchi and T. Fujiwara, Phys. Rev. B72 (2005) 075323). The present ab initio calculation confirms the predicted stepped structure and its bias-dependent STM image. Moreover, two (meta)stable step-edge structures are found and compared. The investigation is carried out also for Ge(111)-2x1 surfaces, so as to construct a common understanding among elements. The present study demonstrates the general importance of the hierarchical research between large-scale and ab initio electronic structure theories.Comment: 5 pages, 4 figures, to appear in Physica

On the simplest sextic fields and related Thue equations

We consider the parametric family of sextic Thue equations $$x^6-2mx^5y-5(m+3)x^4y^2-20x^3y^3+5mx^2y^4+2(m+3)xy^5+y^6=\lambda$$ where $m\in\mathbb{Z}$ is an integer and $\lambda$ is a divisor of $27(m^2+3m+9)$. We show that the only solutions to the equations are the trivial ones with $xy(x+y)(x-y)(x+2y)(2x+y)=0$.Comment: 12 pages, 2 table

Accuracy control in ultra-large-scale electronic structure calculation

Numerical aspects are investigated in ultra-large-scale electronic structure calculation. Accuracy control methods in process (molecular-dynamics) calculation are focused. Flexible control methods are proposed so as to control variational freedoms, automatically at each time step, within the framework of generalized Wannier state theory. The method is demonstrated in silicon cleavage simulation with 10^2-10^5 atoms. The idea is of general importance among process calculations and is also used in Krylov subspace theory, another large-scale-calculation theory.Comment: 8 pages, 3 figures. To appear in J.Phys. Condens. Matter. A preprint PDF file in better graphics is available at http://fujimac.t.u-tokyo.ac.jp/lses/index_e.htm

Birational classification of fields of invariants for groups of order 128128

Let $G$ be a finite group acting on the rational function field $\mathbb{C}(x_g : g\in G)$ by $\mathbb{C}$-automorphisms $h(x_g)=x_{hg}$ for any $g,h\in G$. Noether's problem asks whether the invariant field $\mathbb{C}(G)=k(x_g : g\in G)^G$ is rational (i.e. purely transcendental) over $\mathbb{C}$. Saltman and Bogomolov, respectively, showed that for any prime $p$ there exist groups $G$ of order $p^9$ and of order $p^6$ such that $\mathbb{C}(G)$ is not rational over $\mathbb{C}$ by showing the non-vanishing of the unramified Brauer group: $Br_{nr}(\mathbb{C}(G))\neq 0$. For $p=2$, Chu, Hu, Kang and Prokhorov proved that if $G$ is a 2-group of order $\leq 32$, then $\mathbb{C}(G)$ is rational over $\mathbb{C}$. Chu, Hu, Kang and Kunyavskii showed that if $G$ is of order 64, then $\mathbb{C}(G)$ is rational over $\mathbb{C}$ except for the groups $G$ belonging to the two isoclinism families $\Phi_{13}$ and $\Phi_{16}$. Bogomolov and B\"ohning's theorem claims that if $G_1$ and $G_2$ belong to the same isoclinism family, then $\mathbb{C}(G_1)$ and $\mathbb{C}(G_2)$ are stably $\mathbb{C}$-isomorphic. We investigate the birational classification of $\mathbb{C}(G)$ for groups $G$ of order 128 with $Br_{nr}(\mathbb{C}(G))\neq 0$. Moravec showed that there exist exactly 220 groups $G$ of order 128 with $Br_{nr}(\mathbb{C}(G))\neq 0$ forming 11 isoclinism families $\Phi_j$. We show that if $G_1$ and $G_2$ belong to $\Phi_{16}, \Phi_{31}, \Phi_{37}, \Phi_{39}, \Phi_{43}, \Phi_{58}, \Phi_{60}$ or $\Phi_{80}$ (resp. $\Phi_{106}$ or $\Phi_{114}$), then $\mathbb{C}(G_1)$ and $\mathbb{C}(G_2)$ are stably $\mathbb{C}$-isomorphic with $Br_{nr}(\mathbb{C}(G_i))\simeq C_2$. Explicit structures of non-rational fields $\mathbb{C}(G)$ are given for each cases including also the case $\Phi_{30}$ with $Br_{nr}(\mathbb{C}(G))\simeq C_2\times C_2$.Comment: 31 page

Large-scale electronic-structure theory and nanoscale defects formed in cleavage process of silicon

Several methods are constructed for large-scale electronic structure calculations. Test calculations are carried out with up to 10^7 atoms. As an application, cleavage process of silicon is investigated by molecular dynamics simulation with 10-nm-scale systems. As well as the elementary formation process of the (111)-(2 x 1) surface, we obtain nanoscale defects, that is, step formation and bending of cleavage path into favorite (experimentally observed) planes. These results are consistent to experiments. Moreover, the simulation result predicts an explicit step structure on the cleaved surface, which shows a bias-dependent STM image.Comment: 4 page 4 figures. A PDF file with better graphics is available at http://fujimac.t.u-tokyo.ac.jp/lses

What Happened to Japanese Banks?

This paper argues that the slow and incomplete deregulation of the financial system in the 1980s was the most important factor behind the Japanese banking troubles in the 1990s. The regression analysis of Japanese banks shows that the cross-sectional variation of bad loans ratios is best explained by the variation in the growth of loans to the real estate industry. The variation of growth of real estate lending, in turn, is explained by the varied experience of losing existing customers to capital markets. The rapid appreciation of land prices in the late 1980s also fueled the growth of real estate lending.

Rationality problem for algebraic tori

We give the complete stably rational classification of algebraic tori of dimensions $4$ and $5$ over a field $k$. In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank $4$ and $5$ is given. We show that there exist exactly $487$ (resp. $7$, resp. $216$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $4$, and there exist exactly $3051$ (resp. $25$, resp. $3003$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $5$. We make a procedure to compute a flabby resolution of a $G$-lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a $G$-lattice is invertible (resp. zero) or not. Using the algorithms, we determine all the flabby and coflabby $G$-lattices of rank up to $6$ and verify that they are stably permutation. We also show that the Krull-Schmidt theorem for $G$-lattices holds when the rank $\leq 4$, and fails when the rank is $5$. Indeed, there exist exactly $11$ (resp. $131$) $G$-lattices of rank $5$ (resp. $6$) which are decomposable into two different ranks. Moreover, when the rank is $6$, there exist exactly $18$ $G$-lattices which are decomposable into the same ranks but the direct summands are not isomorphic. We confirm that $H^1(G,F)=0$ for any Bravais group $G$ of dimension $n\leq 6$ where $F$ is the flabby class of the corresponding $G$-lattice of rank $n$. In particular, $H^1(G,F)=0$ for any maximal finite subgroup $G\leq {\rm GL}(n,\mathbb{Z})$ where $n\leq 6$. As an application of the methods developed, some examples of not retract (stably) rational fields over $k$ are given.Comment: To appear in Mem. Amer. Math. Soc., 147 pages, minor typos are correcte